Periodic homogenization for nonlinear integro-differential equations

SIAM Journal on Mathematical Analysis

Volumn 42, Issue 6, 2010, Pages 2652-2680

  Schwab, Russell W ^a  

^a CARNEGIE MELLON UNIVERSITY   (United States)

Author keywords

Homogenization; Jump processes; Levy processes; Nonlocal elliptic equations; Obstacle problem; Optimal stochastic control

Indexed keywords

HOMOGENIZATION; JUMP PROCESS; LEVY PROCESS; NONLOCAL; OBSTACLE PROBLEMS; OPTIMAL STOCHASTIC CONTROL;

GAME THEORY; GEOMETRY; INTEGRODIFFERENTIAL EQUATIONS; MARKOV PROCESSES; NONLINEAR EQUATIONS; OPTIMIZATION; STOCHASTIC CONTROL SYSTEMS; STOCHASTIC SYSTEMS;

PROCESS CONTROL;

EID: 79251483808     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/080737897     Document Type: Article

References (24)

1. A. L. AMADORI, Obstacle problem for nonlinear integro-differential equations arising in option pricing, Ric. Mat., 56 (2007), pp. 1-17.
2. M. ARISAWA, Homogenization of a class of integro-differential equations with Lévy operators, Comm. Partial Differential Equations, 34 (2009), pp. 617-624.
3. M. ARISAWA AND P.-L. LIONS, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), pp. 2187-2217.
4. G. BARLES, E. CHASSEIGNE, AND C. IMBERT, On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana Univ. Math. J., 57 (2008), pp. 213-246.
5. G. BARLES AND C. IMBERT, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), pp. 567-585.
6. R. F. BASS AND M. KASSMANN, Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc., 357 (2005), pp. 837-850.
7. A. BENSOUSSAN, J.-L. LIONS, AND G. PAPANICOLAOU, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978.
8. S. I. BOYARCHENKO AND S. Z. LEVENDORSKIǏ, Perpetual American options under Lévy processes, SIAM J. Control Optim., 40 (2002), pp. 1663-1696.
9. L. A. CAFFARELLI AND X. CABRÉ, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Pub. 43, American Mathematical Society, Providence, RI, 1995.
10. L. A. CAFFARELLI AND L. SILVESTRE, Regularity Results for Nonlocal Equations by Approximation, Arch. Ration. Mech. Anal., to appear.
11. L. A. CAFFARELLI AND L. SILVESTRE, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), pp. 597-638.
12. L. A. CAFFARELLI, P. E. SOUGANIDIS, AND L. WANG, Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. Pure Appl. Math., 58 (2005), pp. 319-361.
13. T. CHAN, Pricing contingent claims on stocks driven by Lévy processes, Ann. Appl. Probab., 9 (1999), pp. 504-528.
14. R. CONT AND P. TANKOV, Financial Modelling with Jump Processes, Chapman & Hall/CRC Financ. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2004.
15. M. G. CRANDALL, H. ISHII, AND P.-L. LIONS, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1-67.
16. B. ENGQUIST AND P. E. SOUGANIDIS, Asymptotic and numerical homogenization, Acta Numer., 17 (2008), pp. 147-190.
17. L. C. EVANS, The perturbed test function method for viscosity solutions of nonlinear PDE, Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), pp. 359-375.
18. L. C. EVANS, Periodic homogenization of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), pp. 245-265.
19. D. A. GOMES AND A. M. OBERMAN, Computing the effective Hamiltonian using a variational approach, SIAM J. Control Optim., 43 (2004), pp. 792-812.
20. P.-L. LIONS and P. E. SOUGANIDIS, Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting, Comm. Pure Appl. Math., 56 (2003), pp. 1501-1524.
21. P.-L. LIONS AND P. E. SOUGANIDIS, Homogenization of "viscous" Hamilton-Jacobi equations in stationary ergodic media, Comm. Partial Differential Equations, 30 (2005), pp. 335-375.
22. B. OKSENDAL AND A. SULEM, Applied Stochastic Control of Jump Diffusions, 2nd ed., Universitext, Springer, Berlin, 2007.
23. H. PHAM, Optimal stopping of controlled jump diffusion processes: A viscosity solution approach, J. Math. Syst. Estim. Control, 8 (1998), article 42281.
24. H. M. SONER, Optimal control of jump-Markov processes and viscosity solutions, in Stochastic Differential Systems, Stochastic Control Theory and Applications (Minneapolis, MN, 1986), IMA Vol. Math. Appl. 10, Springer, New York, 1988, pp. 501-511.